Hybrid methods for direct integration of special third order ordinary differential equations

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Abstract

In this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations (ODEs), y=f(x,y); namely, hybrid methods for solving third order ODEs directly (HMTD). Using the theory of B-series, order of convergence of the HMTD methods is investigated. The main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two-step hybrid method. A three-stage explicit HMTD is constructed. Results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper.

Introduction

Third order ordinary differential equation (ODE) is used in modeling problems arising in various areas of applied science such as biology, quantum mechanics, celestial mechanics and chemical engineering [1], [2]. For instance, the models that describe acoustic wave propagation in relaxing media, draining coating flows e.t.c are all third order differential equations [1], [3], [4], [5], [6], [7], [8]. Some of these equations occur in a special form y=f(x,y(x)), for instance, thin film flow problem studied in [9] and the references therein.

There is always a shortfall in the part of analytical solutions that satisfy most of the ODEs, third order ODEs inclusive. Hence, the search for approximate solutions by numerical means becomes imperative. Over the last few decades, a lot of work has been done on the solutions of third order ODEs, especially in the area of linear multistep related methods. For instance, the P-stable linear multistep method by Awoyemi [10] and the hybrid collocation method by Awoyemi and Idowu [11]. More of these can be found in the works of Majid et al. [12], Olabode and Yusuph [13], Mahrkanoon [14], Guo et al. [5], Myers [6], Abdulmajid et al. [15] , Ken et al. [16] etc. and the references therein.

Traditionally, third order ODEs can be solved by first transforming them into systems of first order equations and applying Runge–Kutta (RK) methods or linear multistep methods, but this could be computationally costlier than the direct methods. In this paper, our main concern is with the initial value problems (IVPs) of special third order ODEs y(x)=f(x,y(x)),y(x0)=y0,y(x0)=y0,y(x0)=y0,where y ∈ Rd, f: R × Rd → Rd is a vector value function. The specialty associated with (1) is the fact that f does not depend on y′, y′′ explicitly. Inspired by Nyström methods (RKN), You and Chen [1] proposed a Runge–Kutta method (RKT) for solving (1) directly. Motivated by two-step hybrid method for solving special second order ODEs proposed by Coleman [17], we propose and investigate three-step hybrid method for solving (1) directly namely, HMTD.

Suppose we want to extend (1.3) of [17] in a way to solve (1) directly, see [18], we get yn+1=2ynyn1+h2i=1sb¯ig(x,y(x))+h3i=1sbif(xn+cih,Yi),Yi=(1+ci)ynciyn1+h2j=1sa¯i,jg(x,y(x))+h3j=1sai,jf(xn+cjh,Yj),where g(y(x))=y and f(y(x))=y. Using Eqs. (4.1) and (5.1) of [17] and applying difference formula, we get Y=12{(c2+3c+2e)yn2(c2+2c)yn1+(c2+c)yn2}+h3(AI)f(xn+ch,Y),yn+1=3(ynyn1)+yn2+h3(bTI)f(xn+chY),in a vector notation for the case y=f(x,y(x)), which is the proposed HMTD method. Where b=[b1,,bm]T,c=[c1,,cm]T,e=[1,,1]T,A=[ai,j]T,Y=[Y1,,Ym]T and I is identity matrix of m × m dimension. The coefficients of the methods are summarized in Table 1.

In Section 2, we present the theory of B-series and the associated rooted trees through which order conditions of the proposed method are derived. Local truncation error and order of convergence of the method are presented in Section 3. We present algebraic order conditions of the method in Section 4. As an example, explicit 3-stage HMTD is presented in Section 5. Numerical experiment is presented in Section 6. And conclusion is presented in Section 7 of the paper.

Section snippets

B3-series and associated rooted trees

It is customary to consider the autonomous case of (1) when working on the order conditions of HMTD methods, as in the case of RK methods, RKN methods and RKT methods for first, second and third order ODEs methods, respectively. y(x)=f(y(x)),y(x0)=y0,y(x0)=y0,y(x0)=y0.Continuous differentiation of the exact solution y(x) with respect to x gives the following: y=y,y=y,y=f(y),yiv=f(y)y,yv=f(y)(y,y)+f(y)y,yvi=f(y)(y,y,y)+3f(y)yy+f(y)f(y),yvii=fiv(y)(y,y,y,y

Local truncation error of HMTD and its convergence order

Like the hybrid methods presented in [17], to derive the order conditions of HMTDs (three-step methods), we consider them as single step methods of the form Gn=Gn1+hΦ(Gn1,h),where Gn is a well defined numerical solution whose initial point G0 is generated by some starting procedure, see [17], [19]. The first part of Eq. (2) can be written as a set of three equations by letting Fn:=(yn+1ynh),so that yn=yn1+hFn1.This implies that Fn=2Fn1+hFn1Fn2+h2(bTI)f(Y).Now, let Un:=(Fn+1Fnh),which

Algebraic order condition

A relationship that exists between the coefficients of a numerical method which causes annihilation of successive terms in a Taylor series expansion of local truncation error of the method is termed order condition of the method [17]. To generate such relationships (order conditions) for HMTD for trees of different orders, Eq. (16) together with (14) and (15) are used. It is worth noting that the ‘order’ referred to here is for the convergence of HMTD not for the order of the rooted trees.

Construction of explicit 3-stage HMTD method

Presented in this section is an explicit 3-stage method of the class of HMTD methods proposed in this paper. To construct a six order method with three stages, equations of order conditions presented in Table 2 up to trees of order seven (see the Theorem in Section 3) together with (21) and s=5, which form system of fourteen algebraic equations in sixteen unknown parameters, are solved. b1=120(c4+2)(c5+2),b2=1105c4c5+5c5+5c4+4(c5+1)(c4+1),b3=12010c4c5+1c4c5,b4=110(2+3c4+c42)c4(c5c4),b5=110c5(c

Numerical experiment

We present in this section numerical experiment conducted by way of applying the new HMTD method derived in this paper alongside some other codes existing in the literature on some test problems in order to evaluate how effective the new HMTD method is. Below is the new method together with the other codes chosen for comparison:

  • HMTD3s6: the 3-stage explicit HMTD method of order six proposed in this paper;

  • RKT3s5: 3-stage explicit RKT method of order five presented in [1];

  • DOPRI5: 7-stage

Conclusion

A family of three-step hybrid methods (HMTD) for solving special third order ODEs directly is presented. This family is similar to the family of two-step hybrid methods for solving special second order ODEs directly [17]. Unlike RKT method [1] , HMTD method has only one equation, which is independent of of first and second derivatives components. Using the theory of B-series with tri-coloured trees presented in [1], order conditions of the HMTD methods are presented. The order conditions are

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